One important concept in geometry is that of a transformation. This is mapping of every point in the plane to a corresponding point in the new plane. There are several different types of transformations that can be considered, including congruent, which is shape and size preserving (e.g. translation), similarity, which is shape preserving (e.g. scaling), and affine, which is collinearity preserving (i.e. preserves parallelism, e.g. rotation). A rotational transformation is affine and congruent.
In finite element analysis, the structural behavior of any body or a region is analyzed by first discretizing the region/domain of interest into several finite elements that are interconnected at nodes. Depending on the type of analysis and accuracy desired, several types of finite elements such as beam, shell and solid elements or a combination of these elements are used for discretizing the domain.
The next step is evaluating the displacements at each node for all finite elements in the region of interest by employing force displacement and equilibrium relations. The nodal displacements are evaluated in terms of translation and rotation components and are often referred as independent degrees of freedom (d.o.f). The nodal displacement for any finite element in a three dimensional space can be expressed as 3 translation components (u, v, w) and 3 rotation components (θx, θy, θz).
Finally, the stresses and strains for each element are evaluated from these nodal translations and rotations using strain displacement and stress strain relations.
Currently in all the available finite element procedures and packages, the following problems exist for structural analysis:
One problem is that beam and shell elements consider both translation and rotation degrees of freedom (DOF), where as solid elements consider only translation DOF. Due to this, DOF mismatch occurs at the shell—solid and beam—solid interfaces which in turn result in wrong results and often singularities during solution.
Another problem is that basic rotational loads like torque and moment are directly related to the rotational DOF and hence solid elements do not consider these basic loads.
Another problem is that translation and rotation DOF at each node are treated as independent of each other which is very difficult to comprehend since from fundamental strain—displacement relations, rotations are dependent on translations and vice versa.
Another problem is that consideration of translation and rotation DOF as independent is even more counterintuitive for cylindrical coordinate (r, θ, z) and spherical coordinate (r, θ, φ) systems. This is because θ direction in cylindrical system, and θ and φ directions in spherical system, represent the rotations themselves. Hence, consideration of rotations about these rotational axes is very confusing.
Current approaches to rotational transformations include the use of solid elements with 6 DOF containing both translation and rotation DOF. However, these elements consume lot of computational resources. Due to this, not all existing analysis systems can support this feature.
Another approach includes the use of multi-point constraints and rigid link elements and special interface elements to tie the shell rotations to solid translations. This is a manual and cumbersome procedure and is extremely difficult to implement in large complex models. Further, the user needs to make several assumptions depending on the application.
All these existing methods find limited use and are employed only to eliminate singularities during solution. Moreover, they still produce inaccurate results at the shell-solid and beam solid interface regions.
There is a need in the art for a system, process, and computer program product for performing improved rotational transformations.